If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
若两数相乘得某数，且任一素数整除该乘积，则该素数必整除两原数之一。
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For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B. For let it not measure A. Now D is prime; therefore A, D are prime to one another.
设两数A、B相乘得C，素数D整除C。假设D不整除A。
[VII. 29] And, as many times as D measures C, so many units let there be in E. Since then D measures C according to the units in E, therefore D by multiplying E has made C.
因D为素数，故A与D互素（VII.29）。设D整除C得E，则D乘E得C（VII.定义15）。
[VII. Def. 15] Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B. Therefore, as D is to A, so is B to E.
又A乘B得C，故D、E之积等于A、B之积。由VII.19得D比A等于B比E。
[VII. 19] But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B. Similarly we can also show that, if D do not measure B, it will measure A.
因D、A互素且为最小（VII.21），由VII.20知D整除B。同理，若D不整除B，则整除A。